Remanence Plots as a Probe of Spin Disorder in Magnetic Nanoparticles

Remanence magnetization plots (e.g., Henkel or δM plots) have been extensively used as a straightforward way to determine the presence and intensity of dipolar and exchange interactions in assemblies of magnetic nanoparticles or single domain grains. Their evaluation is particularly important in functional materials whose performance is strongly affected by the intensity of interparticle interactions, such as patterned recording media and nanostructured permanent magnets, as well as in applications such as hyperthermia and magnetic resonance imaging. Here, we demonstrate that δM plots may be misleading when the nanoparticles do not have a homogeneous internal magnetic configuration. Substantial dips in the δM plots of γ-Fe2O3 nanoparticles isolated by thick SiO2 shells indicate the presence of demagnetizing interactions, usually identified as dipolar interactions. Our results, however, demonstrate that it is the inhomogeneous spin structure of the nanoparticles, as most clearly evidenced by Mossbauer meas...


Introduction
Exchange and dipolar interactions between grains or particles are essential to understanding the behavior of magnetic polycrystalline and colloidal materials. 1,2 Indeed, these interactions are key to the performance of many common magnetic materials, e.g., permanent magnets, 3,4 magnetic recording media, 5,6 magnetically soft materials for high frequency ap-plications, 7 where dipolar interactions may have undesirable effects, such as aggregation of nanoparticles in biomedical applications. [8][9][10] Magnetic interactions control the properties of sufficiently dense assemblies of magnetic nanoparticles and nanostructures, tailoring their functional properties, e.g., blocking (or freezing) temperature, coercivity, remanent magnetization, switching-field distribution and effective anisotropy, among others. [11][12][13][14][15][16] In fact, interactions are the basis of a large number of nanoparticle-based magnetic materials, e.g., superferromagnets, superspin glasses, artificial spin ice, long range self-assemblies, or ferrofluids. 15,[17][18][19][20][21] Given the crucial importance of interactions in magnetic nanostructures, many direct and indirect approaches have been used to try to quantify them: first order reversal curve (FORC) analysis, 22,23 small angle neutron scattering, SANS, 24-27 electron holography, 28,29 magnetic force microscopy, 30,31 Lorentz microscopy, 32 Brillouin light scattering, 33 resonant magnetic x-ray scattering 34 and so on. However, one of the most accepted methods to assess interactions is the remanence plots technique (i.e., Henkel or δM plots), [35][36][37] which is routinely used to evaluate interactions between nanoparticles or grains 38-55 both in fundamental studies 56,57 and in diverse nanoparticle-based applications (e.g., patterned recording media, permanent magnets, or magnetic resonance imaging 11,38,39,55 ). In particular, the δM technique has been used to evaluate dipolar interaction in biomedical applications, where, for example, they have been shown to be crucial for the heating performance of nanoparticles in hyperthermia therapy. [58][59][60][61] Interestingly, this approach is also widely used in paleomagnetism and rock magnetism 62,63 or even biomagnetic systems. 64,65 The technique is based on the fact that, for coherent rotation of non-interacting single-domain particles with uniaxial anisotropy, the isothermal remanent magnetization (IRM, M r ) and the direct current  = 0 has generally been taken as an indication of the absence of interactions. 36,67 It is worth noting that the δM plots for systems of non-interacting particles with cubic anisotropy are intrinsically positive 68 and their shape may vary when different anisotropies co-exist. 69 However, even in the uniaxial anisotropy case, the fact that magnetic single-phase and core/shell nanoparticles (particularly oxide nanoparticles) are usually not monodomains (i.e., cannot be simplified to a system of macro-spins), often exhibiting rather complex internal spin structures, 70-74 may cast some doubts over the validity of the remanence curves approach for the evaluation of dipolar interactions. [44][45][46][47][48][49][50][51][52][53][54][75][76][77] Here we investigate the interactions in γ-Fe 2

Magnetic characterization
To evaluate the strength of interactions as a function of the maghemite concentration, we carried out a systematic δM -plot study across two different series of γ-Fe 2 O 3 nanoparticles: (i) fixed core diameter (d T EM = 8 nm) with a varying-SiO 2 shell-thickness series (VSTx, where x indicates the shell thickness in nm; see Fig. 1) and (ii) a changing-core-diameter (6-11 nm) series with a fixed SiO 2 shell thick enough (≈17 nm) to guarantee the magnetic isolation of the maghemite cores (VCDx, where x indicates the approximate core diameter in nm) (see Methods and Supporting Information). 15,78,79 In all cases, X-ray diffraction and Mössbauer spectroscopy (see Supporting Information) consistently confirmed that the Fe oxide core is maghemite (γ-Fe 2 O 3 ). The same result was previously found for the 8 nm particles using Raman spectroscopy. 80 The temperature dependence of the zero-field-cooled (ZFC) magnetization of all samples shows a peak at low temperatures signalling superparamagnetic (dilute systems) or freezing (dense systems) transitions 15,79 (see Fig. S1a).   Remarkably, a clear dip (≈ 30%) in the δM plot is also observed in sample VST17.
As mentioned, negative δM values are customarily taken to indicate the presence of demagnetizing dipolar interactions. However, the thick SiO 2 shell ensures a large intercore distance yielding an estimated nearest-neighbour dipolar interaction strength, , where d is the average distance between particles and μ their magnetic moment] 86 below 1 K. Therefore, dipolar interactions are negligible in the entire temperature range studied in this work. 86 We can rule out the possible presence of multicore nanoparticles (i.e., the SiO 2 does not coat single γ-Fe 2 O 3 particles but aggregates of particles instead), previously observed in other systems prepared by similar methods, 87,88 as accounting for the observed δM dip: the synthesis procedure was optimized to avoid aggregation as confirmed by a thorough transmission electron microscopy (TEM) study ( Fig. 1), examining thousands of particles. Importantly, the magnetic measurements also confirm that the thick SiO 2 shell effectively suppresses interparticle interactions. For the VSTx series, the dependence of the ZFC peak temperature, T B , on the capping layer thickness indicates that, for sufficiently large t SiO 2 , T B becomes constant (Fig. 3, right axis). This is usually taken to imply negligible dipolar interactions at such great interparticle distances. This is corroborated by the shape of the FC curves, which become flat below T B for interacting particles, while for non-interacting particles they show a monotonic increase below T B (Fig. S1a). Finally, the 7 classical test of H/T scaling (Langevin behavior) was also verified in selected samples with thick silica shells after subtracting a high-field linear dependence (Fig. S1b). In fact, the FC and ZFC magnetization curves (Fig. S1a), the hysteresis loop at 300 K and the ac susceptibility curves measured at different frequencies were simultaneously fitted to an isolated particle model in a consistent way. 15 Additionally, the magnetic volume obtained from a Langevin fit is slightly smaller than the physical volume, again discarding interactions 89 (see Supporting Information). The intensity of the dip of the δM plot depends strongly on the thickness of the nonmagnetic spacer shell. However, the variation of δM with t SiO 2 clearly shows two regimes: strongly decreasing δM for small shell thicknesses, and virtually constant for thick coatings. 8 This behavior indicates that there is more than one mechanism involved, comprising dipolar interparticle interactions (dominating for small t SiO 2 ) and some intraparticle demagnetizing interactions present for all samples, but only evident when dipolar interactions are sufficiently weak. When comparing the shape of the δM plots for strongly interacting VST0 and VSToa samples with the nominally non-interacting VST17 one (Fig. S2), it can be seen that while the non-interacting sample has a rather symmetric peak shape, the interacting samples show a considerably slower approach to zero at high fields (particularly visible when re-scaling the plots -see inset in Fig. S2a-), possibly indicating different mechanisms.
To assess the possible origin of the intraparticle interactions, we studied the low temperature hysteresis loops of the non-interacting samples in the series. As can be seen in Fig. S3, the field-cooled samples exhibit a sizable loop shift, H E , along the field axis (i.e., exchange bias 90 ). Interestingly, all the three samples with large t SiO 2 exhibit roughly the same H E (≈ 100 Oe) value, consistent with the fact that they also show the same δM dip. In fact, the existence of H E can be seen as an "intraparticle" deviation from the Wohlfarth's model for uniform monodomains with coherent switching. Notably, H E has often been observed in oxide ferrimagnets and it is frequently attributed to the presence of a thin highly-anisotropic spin-disordered layer at the surface of the nanoparticle. 70,91 The presence of a spin-disordered layer is also revealed by a reduced "magnetic size" 7.2 nm (extracted from Langevin fits of room temperature loops, see Supporting Information) compared to the particle size evaluated from TEM images d T EM = 8.0 nm. 79 Thus, this spin-disordered layer may be the origin of the unexpected δM dip.
To test this hypothesis we studied the magnetic properties of the VCDx series, comprising nanoparticles with magnetic cores of different diameters coated with a thick silica shell magnetically isolating the cores. The top panels in Figure 74,[96][97][98][99] is unlikely in the present case given the high control of the oxidation process in the particle synthesis; however, it cannot be completely ruled out.
Independently of the origin of the spin canting remaining in the 6 nm nanoparticles, it should be emphasized that it is rather small, equivalent to less than one surface atomic layer. Thus, the Mössbauer results fully confirm that the internal magnetic structure of the nanoparticles is the origin of the δM features in samples without dipolar interactions.

Monte Carlo simulations
To corroborate the results we carried out a systematic Monte Carlo study of nanoparticles with a disordered surface layer. As it can be seen in Fig. 6(a), the simulation of single nanoparticles with such internal magnetic structure clearly shows a sizable δM effect, in concordance with the experimental results, when the surface anisotropy is substantially larger than that of the core. The study of the diverse parameters (exchange, core anisotropy, surface anisotropy and thickness of the disordered layer) evidences that all the parameters are involved in the appearance of negative δM (see Supporting Information). For example, the anisotropies (K Core or K Surf ace ) have a non-monotonic effect on δM . As expected, for sufficiently small K Surf ace (e.g., K Surf ace = 0.1*J F M ), δM virtually vanishes. On the other hand, increasing the thickness of the disordered surface layer leads to larger δM values.
Reducing the exchange causes a decrease in δM in a complex fashion (with multiple peaks).
From the Monte Carlo simulations it can be inferred that the presence of a highlyanisotropic disordered surface layer leads to some locally antiparallel coupled spins. These spins make the magnetization process more difficult (i.e., higher fields are needed for the reversal), which translates into a "demagnetizing" dip in the δM plot. The number of these antiparallel coupled spins (related, e.g., to the disordered surface thickness) and how they interact with each other (changes in J, K Core or K Surf ace ) dictate the strength of the effect. behavior to δM , as further confirmed by the correlation between H E and δM shown in the inset in Fig. 6(b). Since in oxide ferrimagnetic nanoparticles H E is well-known to arise from the surface spin disorder, 100 this implies that the negative δM peak should also be correlated with the presence of surface disordered spins, although a simple one-to-one correlation between both effects cannot be concluded.
The above results demonstrate that using remanence plots to establish and quantify dipolar interactions in nanoparticle systems may be misleading, since systems with an absence of interparticle dipolar interactions but with an inhomogeneous internal magnetic structure (which does not behave as a simple macrospin) present clear non-zero δM signals. More careful analysis of the shape of the δM dips (see Supporting Information) indicates that when the dipolar interactions are strong the shape of the δM is more asymmetrical and δM goes to zero at much higher fields than when δM arises from internal exchange interactions only.
The measurement of a series of samples has allowed meaningful conclusions to be reached in the present study. However, it would be very difficult to ascertain the contribution of interparticle dipolar interactions to the δM curve of a single particle assembly alone.

First Order Reversal Curves
The consistency of the above results justifies the use of our samples as benchmark systems

Conclusions
Inhomogeneous intraparticle magnetic structure has been clearly shown to cause the appearance of a strong δM dip in isolated maghemite particles (for which TEM and magnetic behavior consistently confirm magnetic isolation of the maghemite cores by thick silica coating shells), very similar to the dips customarily attributed to dipolar interactions. The exchange coupling of a highly anisotropic layer of disordered surface spins (as confirmed by Mössbauer spectrometry) with a uniform ferrimagnetic core is the mechanism responsible for both exchange bias and the non-zero δM , as indicated by the correlation between the two parameters. The replication of the observed δM dips in Monte Carlo simulations of isolated particles with an ordered core/disordered surface magnetic structure provides further support for the proposed intraparticle exchange mechanism. Our work thus strikes a cautionary note on the widespread use of remanence plots to assess interparticle/intergrain interactions, since effects due to complex internal magnetic structure may be easily overlooked, particularly in oxide particles, where surface spin disorder is commonly found. Conversely, our results demonstrate that, provided dipolar interactions can be safely ruled out by other means, δM plots can be used to sensitively detect and possibly quantify intraparticle exchange coupling in magnetically non-uniform nanoparticles.

Methods
Two series of samples were prepared:  were collected by centrifugation. The nanoparticles can be dispersed in n-hexane and then precipitated again by adding acetone to remove the excess oleic acid. To obtain smaller maghemite nanoparticles, the above procedure was repeated by using a reduced amount of oleic acid. As the amount of oleic acid is decreased, the decomposition temperature of iron precursors decreases. The lower decomposition temperature probably leads to a larger number of seeds, yielding smaller nanoparticles. Magnetization reversal characteristics of the samples were also studied using the FORC method. 22,106-108 A strong magnetic field is applied to first saturate the sample. The field is then lowered to a specified reversal field H r , and the magnetization is measured as H is increased back to saturation. A series of these measurements at various reversal fields makes up a complete set of FORCs. The FORC distribution is then determined using a mixed partial derivative, 109

Silica coating of the maghemite nanoparticles
Alternatively, the FORC distribution can also be expressed in terms of the local coercivity to be equal to that of the core.

ASSOCIATED CONTENT Supporting Information
Additional information on the δM and the in-field Mössbauer techniques, table with the complete results of the Mössbauer spectra fits, details of the Monte Carlo simulations, FC and ZFC magnetization curves of the VST series (Fig. S1a), Langevin scaling of M(H;T) data measured in VST45 (Fig. S1b), details on the estimate of the "magnetic size" from Langevin fits, δM plots of all the VST series and graphical analysis of the intraparticle and interparticle contributions to the dip (Fig. S2), example of hysteresis loops measured after ZFC and FC (for sample VST17, Fig. S3); X-ray diffraction patterns and lattice parameter across of the maghemite cores of different size (Fig. S4); complete results from Monte Carlo simulations showing the dependence of δM on core anisotropy (Fig. S5), surface anisotropy ( Fig. S6), exchange coupling constant (Fig. S7) and disordered surface thickness (Fig. S8).
This material is available free of charge via the Internet at http://pubs.acs.org.  Table S1. Neither sample contains Fe 2+ ions, the isomer shift and the hyperfine field being typical of maghemite nanoparticles 6 . It is also interesting to observe a significant increase of

AUTHOR INFORMATION
Fe at the expense of Fe populations, suggesting an excess of octahedral Fe sites at the surface of maghemite nanoparticles.
Note that the in-field spectra in Figure 5 show that the second and fifth lines have a nonzero intensity. Usually, when these peaks are distinctly observed, they evidence a canted structure for Fe 3+ magnetic moments with respect to the applied field (non-collinear magnetic structure) 7,8 . For a thin absorber the relative area of the six lines is given by 3 As shown in Figure 5 and

III. Monte Carlo Simulations
We considered isolated ferrimagnetic (FiM) spherical nanoparticles (consisting of a core and a surface) of diameter d, expressed in lattice spacings, on a simple cubic lattice. We take into account explicitly the microstructure of the system at an atomic scale. The spins of the nanoparticles are located at each lattice site of the core and the surface. They interact with nearest neighbors through Heisenberg exchange interactions and at each crystal site they experience a uniaxial anisotropy. At the surface of the particles, the crystal symmetry is reduced and consequently the anisotropy is stronger than in the core 9,10 .
In the presence of an external magnetic field, the total energy of the system is

IV. Langevin Fit
To further corroborate the non-interaction behaviour, we performed one of the classical tests of ideal superparamagnetic behaviour, i.e., the H/T scaling in the Langevin response above TB. As shown in Figure S1(b), the loops measured at 200, 250 and 300 K nicely scale when plotted as a function of H/T. Moreover, the data measured in the different VCDx samples is well fitted by a simple Langevin function 12 with the derived nanoparticle magnetic diameter (e.g., dmag = 7.2 nm for VCD8) systematically slightly smaller than the diameter obtained from TEM (e.g., dTEM = 8.0 nm for VCD8) (Ref. 12). If interactions were present, a magnetic volume larger than the physical volume would be expected 13 . The fact that the "magnetic size" is slightly smaller than the TEM size may be also taken as an indication of a magnetically disordered surface layer; in fact, sample VCD6a shows the smallest difference between the two diameters in all the VCD series 12 .

V.
δM comparison in the VST series Figure S2. (a) δM curves measured in the VST series (the particles with silica shells 45 and 62 nm thick showed curves almost identical to that of the VST17 sample). The inset shows the data of three representative samples with both axes normalized in order to highlight shape differences. (b) The data of the VST0 (with inter-and intra-particle interactions) and VST17 (only intraparticle interactions present) samples is shown together with their difference curve, highlighting the contribution of dipolar interactions in VST0. Figure S3. Hysteresis loops measured in sample VST17 (VCD8) at 5 K after zero-field-(ZFC) and field-cooling (FC) from room temperature. The latter shows a horizontal shift of HE = 110 Oe. The coercivity is 540 Oe in both cases.

VI. Exchange Bias
VII. X-ray diffraction Figure S4. X-ray diffraction patterns from iron oxide cores of the VCD sample series; the values in the legend are the corresponding TEM diameters in nanometers. The data have been background corrected. The solid line curves are fits obtained using profile-matching based on the allowed Bragg reflections for maghemite over the measured 2θ range; i.e., from left to right, the (2 2 0), (3 1 1), (4 0 0), (4 2 2) and (5 1 1). The resulting (fitted) lattice parameter values are presented in the inset, where the errors bars are smaller than the plotted points. The lattice parameter at dTEM = 8 nm is taken from a similar analysis. 14